Numbers Divisible by 9

Table below shows a list of multiples of 9 and the sum of their digits. We realised that, the sum of the digits is also multiple of 9.

Multiples of 9	Sum of the digits
9	9
18	1 + 8 = 9
27	2 + 7 = 9
36	3 + 6 = 9
45	4 + 5 = 9
54	5 + 4 = 9
63	6 + 3 = 9
72	7 + 2 = 9
81	8 + 1 = 9
90	9 + 0 = 9
99	9 + 9 = 18	1 + 8 = 9
108	1 + 0 + 8 = 9
117	1 + 1 + 7 = 9
126	1 + 2 + 6 = 9
135	1 + 3 + 5 = 9
144	1 + 4 + 4 = 9
153	1 + 5 + 3 = 9
162	1 + 6 + 2 = 9
171	1 + 7 + 1 = 9
180	1 + 8 + 0 = 9
189	1 + 8 + 9 = 18	1 + 8 = 9
198	1 + 9 + 8 = 18	1 + 8 = 9
207	2 + 0 + 7 = 9
216	2 + 1 + 6 = 9
225	2 + 2 + 5 = 9
234	2 + 3 + 4 = 9
243	2 + 4 + 3 = 9
252	2 + 5 + 2 = 9
261	2 + 6 + 1 = 9

Therefore, we can conclude that

If the sum of the digits of a number is divisible by 9, the number must also divisible by 9

or

If the sum of the digits of a number is a multiple of 9, the number must also a multiple of 9.

For example:
Is 972635 a multiple of 9?

Sum of the digits:
9 + 7 + 2 + 6 + 3 + 5 = 32

32 is not multiple of 9, hence 972635 is not a multiple of 9.

Let's see another example:
Is 683746533 divisible by 9?

Sum of the digits:
6 + 8 + 3 + 7 + 4 + 6 + 5 + 3 + 3 = 45

45 is divisible by 9, hence 683746533 is divisible by 9